First Extension: Random Weights

In this chapter a Law of Large Numbers is proved for an extension of the functionals

V

n

(

f

,

X

) and

V

′

n

(

f

,

X

). For the first one, the summands

$f(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})$

are replaced by

$F(\omega ,(i-1) \varDelta _{n},X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})$

for a function

F

on

Ω

×ℝ

+

×ℝ

d

, where

d

is the dimension of

X

, and likewise for the functional

V

′

n

(

f

,

X

). The results are perhaps obvious generalizations of those of Chap.

3

, the main difficulty being to establish the assumptions on

F

ensuring the convergence.

The motivation for this is to solve parametric statistical problems for discretely observed processes: in the last section one considers the solution of a (continuous) stochastic differential equation whose diffusion coefficient depends smoothly on a parameter

θ

. Then, on the basis of discrete observation along a regular grid, one shows how the previous Laws of Large Numbers can be put to use for constructing estimators of

θ

which are consistent, as the discretization mesh goes to 0.